Hypoid gearing



Nov. 29, 1960 E. WILDHABER 2,961,888 I HYPOID GEARING Filed June 11.1958 4 Sheets-Sheet l J 47 i INVENTOR;

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Nov; 29, 1960 E. WILDHABER HYPOID GEARING 4 Sheets-Sheet 2 Filed June11; 1958 INVENTOR.

FIG/l Nov. 29, 1960 United States Patent HYPOID GEARING ErnestWildhaber, Brighton, N.Y. (124 Summit'Drive, Rochester 20, N.Y.)

Filed June 11, 1958, Ser. No. 741,280 17 Claims. 01. 74-4595 The presentinvention relates to hypoid gearing, having angularly disposed andoffset axes, wher'eat least one member of the gear pair has spirallyarranged teeth adapted for successive engagement along the tooth length.Particularly it relates to a twisted or warped tooth shape such that thelines of instantaneous tooth contact include relatively large angleswith the lengthwise'direction of the teeth, and such that an intimatetooth contact is att ained.

Tooth shapes of this general character are disclosed in my Patent No.1,816,272, granted July 28, 1931, and furtherin my pending patentapplications entitled Gearing, filed November 1, 1955, and HypoidGearing, filed May 8, 1958, Serial Nos. 544,270 and 733,990,respectively.

In the tooth shapes referred to, the intimacy of tooth contact ofconjugate tooth surfaces varies'along the length of the teeth and ismost intimate at one end, opposite tooth sides having their mostintimate contact at opposite ends of the teeth.

One object of the invention is to so shape the teeth that they can beused up to the very point where the curvatures of mating tooth surfacesare fully matched, so that in all sections through that point the convexand concave intersection curves of mating tooth surfaces have equalcurvature radii, and almost surface contact is achieved at that endpoint. A related object is to increase the load capacity of such teethand their usable face width.

A further object is to so shape the teeth of a hypoid gear pairconsisting of a gear and a pinion, to provide such a tooth shapethereon, that the most intimate tooth contact exists not only at an endpoint but along the whole end profile of a gear tooth.

A still other aim is to provide teeth of the last-named character thathave constant profiles from end to end of the teeth on at least the gearmember, and teeth that have constant profiles on both the gear and thepinion and that can be produced by form cutting in the manner outlinedhereafter. A related aim is to provide mating tooth surfaces that meshalong a helical surface of action.

A further object is to devise teeth of the character referred to, whoseside tooth surfaces are composed of straight profiles on both the gearand the pinion, and to so form the ends of the gear teeth that themaximum intimacy of contact is had all along the end profiles of thegear teeth.

An alternative aim is to provide standard ends on the gear teeth and toprovide constant and'moderately curved profiles on the gear and piniontooth sides, so that maximum intimacy of contact is attained all alongthe end profiles of the gear teeth.

A further object is to provide pinionteeth overlapping in length theends of the gear teeth and to relieve the ends of the pinion teeth onthe side where the tooth contact is most intimate, relief starting atthe line that en gages the end profile of the gear teeth, to rendertooth contact with matched curvatures possible without interference.

I 2,961,888 Patented Nov. 29, 1960 Also the production of 'such gearingshall be improved.

Other objects will appear in the courseof the specification and in therecital of the appended claims. These objects may be attained singly orin any combination.

In the drawings:

Fig. 1 is a diagramamtic view of a hypoid gear pair shown chiefly by itspitch surfaces and constructed according to the present invention, theview being taken along the gear axis. Fig. l is further an explanatorydiagram, and also shows the helical surface of action of this gear pair.

Fig. 2 is a view and diagram corresponding to Fig. 1, the view beingtaken in the direction of the center line of the gear pair. I

Fig. 3 is a view of a hypoid gear pair corresponding to Figures 1 and 2,and taken in direction of the center line of the gear pair, the gearmember being shown in axial section.

Fig. 4 is a mean normal section taken through a few teeth of this gearpair, through mean point 41.

Fig. 5 is a fragmentary View showing a tooth space of the gear 36 ofFig. 3, taken at right angles to its pitch surface.

Fig. 6 is a diagrammatic view corresponding to Figures 1 and 2, taken atright angles to a plane containing the axis of the helical surface ofaction and the center line of the gear pair.

Figures 7 and 8 are diagrams corresponding to Fig. 6, looking along theaxis of the helical surface of action, and referring to opposite sidesof the teeth respectively.

Fig. 9 is a diagrammatic view of a hypoid gear pair constructedaccording to a modification, taken along the center line of the gearpair, the gear being shown in axial section.

Fig. 1O is a fragmentary mean normal section through meshing teeth ofthe gear pair shown in Fig. 9, at a larger scale.

Fig. 11 is a diagrammatic view and section similar to Fig. 9 butreferring to a further modification.

Figures 12 to 18 are enlarged sectional views corresponding to theembodiment of Figures 1 to 8. Figures 12 and 13 are fragmentaryperipheral sections of the hypoid gear, taken along lines 12-42 and 1313respectively. Fig. 14 is an axial section of the hypoid gear, showingopposite sides of the teeth. Fig. 15 is a fragmentary section takenalong the pitch surface of the gear and viewed at right angles to thepitch surface. Fig. 17 is an axial section of the mating pinion, showingopposite tooth sides, interfering teeth being omitted for convenience.Fig. 18 is a sectional view of a pinion tooth, taken along the pitchsurface, its central pitch line being developed into a plane.

Figures 19 and 20 are diagrams illustrating a way of rough cuttingopposite sides of the gear teeth.

Fig. 21 is a normal section taken through a tooth surface of the gear,showing also a finish-cutting tool in engagement with one side thereof.

Fig. 22 is a side view of the tool shown in Fig. 21.

Fig. 23 is an end view of this tool, looking at the cutting portion.

Figures 24 to 27 are diagrams illustrative of a way of rough-cutting thepinion, a fragmentary normal section through a tooth space being shown.Cross-hatching is omitted to better show the dotted lines.

.Fig. 28 is a diagrammatic and fragmentary view of a gear pitch surface,taken at right angles thereto, and Fig. 29 is a corresponding frontelevational view, both figures illustrating a modified form ofproduction resulting in 'a somewhat modified "form of'teeth.

Figures 1 to 5"show a tape'redgear 30 in engagement with a' pinion"31T'Th'e gear and'piiiion are rotatably mounted on angularly disposedand offset axes 32, 33 respectively, here shown at right angles to eachother. Numeral 34 denotes the center line of the gear pair, intersectedat right angles by the axes 32, 33.

The pitch surface 30 of the gear 30 has a convex contour 35, that is aconvex profile in axial section. The mating pitch surface 31 of thepinion 31 contains a concave contour 36 such that the two pitch surfaces30, 31 contact along a line 37 (Fig. l).

' Pitch surfaces of this kind and teeth built around them have beendisclosed in my companion application Serial No. 733,990. They have anincreased duration of pitch line contact and more teeth in simultaneouscontact than conventional designs.

Some gear teeth are shown in section at 29 in Fig. l, the section beingtaken along the pitch surface 30. Some tooth tops 47 are also shown, anda conical section 47' through opposite ends of the convex tooth tops.The section is seen to bulge out between the tooth ends.

In a mathematically exact embodiment without easeoff mating toothsurfaces have constant matching profiles and are such as can be tracedon the rotating gear and pinion by a line describing a helical surface38 that extends at a constant lead and at a constant distance from anaxis 40. The helical path 37 of mean point 41 of the describing line isthe contact line of the pitch surfaces 30', 31' and the mean path ofcontact. The mating teeth contact along the describing line (57, 58respectively) in all turning positions.

To provide these conditions, the position of axis 40, the lead along itand the turning ratio about it have to conform to the axis, lead andturning ratio of a basic helical member of the hypoid gear pair. Basichelical members are known. They have the same kind of relative motionwith respect to the hypoid pair as the members of the hypoid pairthemselves. This motion can at any instant be considered a helicalmotion about an instantaneous axis. This motion should not only be aboutthe same axis, but also should have the same lead. Reference is made tothe companion application Serial No. 733,990 for more detailedinformation. An outline shall be given here however.

Axis 40 should lie in a plane parallel to the axes 32, 33. Its directionis assumed to provide a suitable inclination of the path of contact 37.

Next the direction of the said instantaneous axis is determined as ifthe gears 30, 31 were bevel gears with intersecting axes parallel to theaxes 32, 33 and having the given tooth ratio. The turning ratio aboutaxis 40 can be determined as if for a bevel gear whose axis is parallelto axis 40 and passes through the intersection point of the axes of saidbevel gears, having the same instantaneous axis as these. This providesthe turning motion for any assumed direction of axis 40.

Axis 40 furthermore should intersect the center line 34, and should havea definite distance E =4344 (Fig. 1) from the pinion axis 33, dependingon the inclination i (Fig. 2) of axis 40 to the direction of the pinionaxis 33. When E denotes the offset of the axes 32, 33, E and the lead Labout axis 40 can be computed with the formulas inclinations, whichincrease from the outer end 50m the inner end 51 on the longitudinallyconvex side 48 of the teeth, and decrease on the opposite side 49. Thechange of profile inclination of the normal sectional profiles is atleast twenty degrees from end to end of the teeth.

The difference of the average inclination of these profiles on theopposite sides 48, 49 should depend on the limit pressure angle: Anormal plane laid through mean point 41 at right angles to the pitchline and to the tooth direction intersects the center line 34 at a point52 (Fig. l). The connecting line 41-52 is what I have called the limitnormal. The two side surfaces 48, 49 should be about equally inclined tothe limit normal when they pass through mean point 41.

A further meaning of the limit normal is illustrated with Fig. 4, whichis a normal section through teeth contacting at 41. It shows the path 55along which the pitch line of the pinion approaches the pitch ,point 41of the gear and recedes from it. In the immediate vicinity of the pitchpoint (41) the relative path is in a direction 56 that is inclined atthe limit pressure angle to the pitch vertical 45. The limit pressureangle is understood to be the inclination of the limit normal 41-52 tothe pitch surfaces at mean point 41, or to their tangent plane.

The inclined direction (56) of approach is further described in ArticleV of my eight articles on the Basic Relationship of Hypoid Gears thatwere published in American Machinist in 1946, see Fig. 32, thereof. Inthis respect hypoid gears differ from bevel gears with intersectingaxes, where the last approach is at right angles to the pitch surface.This will be further referred to.

Surface of action In principle there is a surface of action for each ofthe two sides of the teeth, both surfaces containing path 37. Inconventional gearing the surfaces of action of opposite tooth sidesintersect and include a substantial angle with each other. If this isfollowed here, the outside ends of the two helical surfaces of actioncontain coaxial spaced helices. This generally results in root surfacesthat are differently tapered on opposite sides of the teeth. That is,the ends of the generating lines of opposite tooth sides describe rootlines of different taper. It is desirable to have a common root surfacefor both tooth sides, that is a surface of revolution. This isattainable in accordance with the invention by providing a pair ofsurfaces of action which nearly coincide, and which are tangent to eachother along the mean path of contact 37.

In Fig. l numeral 57 denotes the describing line or generating line thatdescribes the longitudinally convex side 48 of the gear teeth and themating tooth sides of the pinion. It is a straight line in thisembodiment. The straight line 58 is the describing line for thelongitudinally concave side of the gear teeth and the matinglongitudinally convex side of the pinion teeth.

Line 57 is determined when it passes through the outer end point 53 ofthe path of contact 37. It should then have a pressure angle equal tothe limit pressure angle at that point. That is, it should lie in aplane perpendicular to the limit normal.

The limit normal The limit normal is the connecting line of point 53with a point 60 (Figs. 1, 6, 7) of the center line 34. The distanceB=4460 of point 60 from the pinion axis 33 will now be computed. Likeany other contact normal, the limit normal fulfills the condition thatits leverages with respect to the two axes are in the proportion of therespective tooth numbers. In other words, a force directed along thisnormal should exert turning moments on the two members of the gear pairin the proportion of their tooth numbers.

In the view along center line 34, Fig. 2, the limit normal 53-60includes an angle k with the direction of the pinion axis 33. At point60 the said force can be. resolved into two components. center line 34,and the other lies in a plane perpendicular to center line 34 andincludes the above said angle k with the direction of the pinion axis.The component along center line 34 intersects both axes 32, 33 andtherefore exerts no turning moment on either member. The other componentexerts a turning moment on the gearproportional to the horizontalcomponent F cos k of the force F and. to the distance (E+B), that is Fcos k(E+B). It exerts a turning moment F(sin k)B on the pinion. The twomoments should be in the proportion M=Nln of their tooth numbers, Nbeing the gear tooth number. Hence (E+B) cos k=MB sin k throughtransformation:

B- M tan lc-l The limit normal 53-60 can now be plotted in Figures 6 and7.

Hitherto it has been considered impossible to have conjugate toothaction up to points whose tooth surface normal coincides with the limitnormal. I have found that with the measures disclosed hereafter it ispossible to have tooth action up to these very points. Not only doesthis extend the duration of contact of such warped teeth, but it securesthe most intimate contact. 7 At these points the curvatures ofcontacting tooth surfaces are matched completely, so that the contactachieved is almost surface contact.

The describing line To this end, the describing line 57 or at least itstangent at point 53 should lie in a plane perpendicular to the limitnormal 53-60. Line 57 or its tangent also lies in the tangent plane ofthe assumed helical surface of action. This helical surface may coincidewith the helical surface containing the pitch vertical 45. Thus thedescribing line or its tangent can be determined as the intersectionline of said tangent plane and the plane perpendicular to the limitnormal.

We may first consider the said tangent plane in the position where itpasses through mean point 41.: It contains the pitch vertical 45 and thehelix tangent at point 41, that is the tangent to path 37. Its trace 61'(Fig. 7) on a plane 62 (Fig. 6) through point 41 can readily bedetermined with known procedure. Plane 62 and plane 63 are both parallelto the drawing plane of Fig. 7 and perpendicular to axis 40. The trace61" with plane 63 is parallel to trace 61" and its distance therefromcorresponds to the distance 64 (Fig. 6) of the planes 62, 63. Thetangent plane of the helical surface of action at end point 53 has atrace 61 on plane 62. This plane has the said distance 64 from point 53.Trace 61 (Fig. 7) has the same distance from axis 40 as trace 61" and adifferent angular position. With respect to trace61 it is tu-rnedthrough the same angle 41-40-53 (Fig. 8) as end point 53 is turned frommean point 41.

Next we consider the plane through 53 perpendicular to the limit normal53-60 and determine its trace 65 with the same plane (62). Point 53 hasa distance 66 from the drawing plane of Fig. 7 which is perpendicular toaxis 40 and contains center line 34. In Fig. 6 this distance appears asthe distance of point 53 from center line 34. We may plot distance 66 inFig. 7 on a line 60-60 perpendicular to the projected limit normal53-66, draw line 60"-53 and a line 53-67 perpendicular thereto. Line 68is drawn parallel to projected line 60-53 at the aforesaid distance 64therefrom. Itintersects line 53-67 at point 67. The sought trace 65passes through point 67 and is perpendicular to the projected limitnormal 53-60. The two traces 61 and 65 inter- One of these extends alongsect at a point 70. The describing'lin'e 57 or its tangent is a portionof the connecting line 53-70.

Tooth surfaces 48of the gear'and the mating tooth surfaces of the pinionare described by moving line 57 helically about and along axis 40 at thelead L while the gears turn on their axes 32, 33 as if they would runtogether. The turning motion about axis 40 thereby should conform to theturning motions of the gear pair in the above described manner. Line 57also traces the helical surface of action in space. V

The describing line 58 for the opposite side of the teeth is determinedat the inner end position, at point 54 of the path of contact 37. Theprocedure is analogous to the one described. First the limit normal atpoint 54 is determined in Fig. 2 as the connecting line 34-54, to obtainanother angle k. Then a new distance B is computed with the givenformula. Distance B may then be plotted in Fig. 6 and in Fig. 8 from thepinion axis on center line 34 to obtain the intersection point 601' ofthe limit normal 54-60i.

Again we determine the traces in a plane parallel to the drawing planeof Figures 7 and 8 and having the distance 64 from point 54. The trace611i of the same tangent plane has the same distance from axis 46 astrace 61" of Fig. 7, and is merely turned with respect to it. The trace651' of the plane perpendicular to the limitnormal 54-6tii passesthrough a point 67i. To determine this point, a distance 66i is plottedin Fig. 8 on a line 6tli-6tli drawn at right angles to the projectedlimit normal, to obtain a point 601'". Distance 661' equals theprojected distance of point 54 from center line 34' in Fig. 6. Line54-601 is drawn and through point 54 a line 54-67i perpendicularthereto. Line 681' is drawn parallel to the projected limit normal54-60i at the distance 64 (Fig. 6) therefrom. Its intersection with line54-67i is the sought point 67i. Trace 65i is drawn through point 67i atright angles tothe projected limit normal 54-6tli. The two traces 651'and 61i intersect at 7 01'. The tangent to the describing line 58 or theline itself is part of the connecting line 54-70i.

As line 58 is moved helically about axis 40, at the prescribed ratio andlead,,it describes side 4? of the gear teeth and the mating toothsurfaces of the pinion on the rotating gear and pinion, and in spacedescribes the surface of action.

-So far fully conjugate tooth surfaces have been described, where thetoo-th bearing sweeps the entire length of the gear teeth in the givenrunning position. It is desirable and customary to slightly ease off thetooth ends on one or both members of the gear pair, to render the gearsless sensitive to deflection under load and to slight errors inalignment and manufacture.

This may be attained by slight departures from the mathematically exactprocedures to be described, such departures being customary in the art.It should be understood that the description of the exact form isintended to include the slight departures of intended ease-off. Thedescribing profiles themselves may be slightly mismatched. Thus forinstance a straight cutting edge may be used on one member and aslightly concave one on the mate, but we still call them matchingprofiles.

The gearing is also applicable to gear pairs whose shaft angle differshome right angle. In such cases the limit normal no longer intersectsthe center line of the gear pair. It may be determined in accordancewith my aforesaid articles. At any point (53 or 54) the tangent plane ofthe contacting pitch surfaces is first determined. It is referred to asthe pitch plane. The inclinations of the gear and pinion axes from thisplane are referred to as the pitch angles, and the distances of theirintersections with this pitch plane from the said point are referred toas the cone distances. The limit normal is perpendicular to the pitchlines contacting at said point, and its inclination to the pitch planeis referred to as the limit pressure angle. Itcan be determined withFormula 15 given in Article II.

Tooth ends of gear In accordance with one aspect of the invention thedescribing lines 57, 58 are straight lines, and the tooth ends 50, 51 ofthe gear are so Shaped or dimensioned that the curvatures of contactingteeth are completely matched, or closely and equally matched, along awhole gear end-profile, when its points are in contact position. On thelongitudinally convex sides 48 of the gear teeth they are matched on theend profile that passes through point 53 at the outer end. On thelongitudinally concave sides 49 of the gear they are matched on theinner end profile that passes through point 54.

The tangents may be substituted for the profiles of the end surfaces sodetermined. The surfaces 50,- 51 are then conical surfaces coaxial withthe gear. Their straight profiles in axial sections are seen to convergein a direction from tooth bottom to tooth top, at an angle larger thanthirty degrees, so that the face width at the tooth tops 47 is smallerthan at the tooth bottom 46.

It will now be shown how the inclination of the tooth ends 50, 51 may bedetermined.

Planes are considered that are perpendicular to the describing line, andtheir traces are determined in the drawing plane of Figs. 7 and 8, whichplane contains the center line 34 and is perpendicular to axis 40. Theconsidered planes have traces at right angles to the describing line.Such a plane at point 53 (Fig. 7) has a trace 71 that intersects thecenter line 34 at 60, because this plane contains the limit normalpassing through point 60. The plane normal to the describing line 57through point 53' thereof has a trace 71' parallel to trace 71.

The limit normal at point 53' can be directly drawn in the view Fig. 2,and determines the angle k used for computing the distance B. Knowingthe new B we can plot the point 60 (Fig. 7) where the new limit normalintersects the center line 34. As trace 71' is seen to be offset frompoint 69 the surface normal of the teeth contacting at point 53 does notcoincide with the limit normal 53'60' at that point. To make it coincidea helical displacement of point 53' in counter-clockwise direction isneeded. A helical displacement about axis .40 may be assumed; and theabove procedure is then repeated until the position 53 of point 53' isfound, where the surface normal coincides with the limit normal. The sodetermined point 53" is a point of the gear outside surface 50. Otherpoints may be similarly determined. Surface 50 also contains point 53.

While I have described chiefly geometrical procedures, the determinationcan of course also be made by computation, which expresses the describedprocedure. In this case point 53' is preferably assumed at aninfinitesimal distance from point 53. The required inclination of theconical end surfaces is computed with the known procedures ofmathematics dealing with infinitesimal changes.

On the opposite side of the teeth, the plane perpendicular to thedescribing line 58 at point 54 intersects the drawing plane in a trace72 (Fig. 8) that passes through point 60i. The trace 72' of the planeperpendicular to the describing line at point 54 thereof is parallel totrace 72. The limit normal at point 54' is determined with the abovedescribed procedure. It intersects the center line 34 at a point 60i'offset from trace 72'. A small clockwise helical motion of point 54about axis 40 is here required. At a position 54" the limit normal therecoincides with the normal of the contacting tooth surfaces. The insidesurface 51 is the surface that contains point 54" in addition to point54.

The tooth ends 50, 51 may also be determined experimentally by trial.

Modifications The embodiment illustrated in Fig. 9 differs from thedescribed embodiment merely in the shape of its end surfaces 50', 51'and in the curvature of the describing lines. Everything else is thesame as described, including the tangents of the describing lines attheir mean point (41) and the helical mean path of contact 37.Conventional end surfaces 50', 51 are used, that have parallel straightprofiles. However any suitable shape may be assumed if desired.

Instead of determining the inclination of the end surfaces from assumeddescribing lines and tooth profiles, the describing lines and toothprofiles are here determined from the assumed shape of the surfaces 50',51'.

We consider point 53' of Fig. 7 and its limit normal 53'60'. The normalto the tooth surfaces contacting at 53 depends on the direction of thedescribing line at that point. By curving the describing line 157 wecould bring this normal into coincidence with its limit normal 53'60'.Then it should be so curved that its tangent at 53' lies in a planeperpendicular to the limit normal 53-60'. The profile curvature is bestdetermined by computation, assuming an infinitesimal distance 53-53.

When line 157 is curved in a plane parallel to axis 40, so that in Fig.7 the tangent at 53 in projection coincides with the tangent 57 at point53, then the trace of the plane normal to profile 157 at 53' remainsparallel to trace 71' and is shifted laterally. The curvature should besuch that the shifted trace passes through point 60. This requires aconvex profile 157 on the gear, and a matching concave one on thepinion.

Instead of determining the curvature required to make point 53' an endpoint of tooth action, the helix passing through 53' is determined, andits intersection with the outside surface 50' of the gear. This meansdisplacing point 53 in clockwise direction in Fig. 7. The describedconstruction is then repeated for this point. It results in a profile157 more convex on the gear than if 53', as shown, were the end point.

The opposite side of the teeth is similarly treated. A slightly convexprofile on the gear causes the surface normal at point 54' to passthrough point 60i and to coincide with the limit normal 5460i. Theintersection point of the helix through 54 with end surface 51' isdisplaced in counter-clockwise direction from point 54. When theconstruction is repeated for this intersection point a convex profile158 of more curvature results.

The tooth profiles of a mean normal section of the teeth are shown inFig. 10, at an enlarged scale as compared with Fig. 9. This section isunderstood to be at right angles to the tooth direction. The gear hasconvex profiles 74, and the pinion has concave profiles 75. They areshown in relation to the pitch vertical 45.

A further embodiment is illustrated in Fig. 11. Here the axis 240 of thebasic helical member is not at a distance from the tooth zone, but lieson the root surface of the gear 230. It can be considered the line ofcontact of mating pitch surfaces and a path of contact.

The describing lines 257, 258 are shown at mean point 241. Thedescribing line 257 is further shown in a position where it passesthrough the outer end point 253 of the path of contact, where itsdirection is determined in the manner described. Line 257 corresponds tothe longitudinally convex side of the gear teeth and to the mating sideof the pinion teeth. The opposite describing line 258 is further shownpassing through the inner end point 254 of the path of contact.

Axis 240 intersects the center line of the gear pair. Its inclination(i) and its offset (E,.) from the pinion axis 33, as well as the lead(L,) are determined in exactly the same way as in the previouslydescribed embodiments. The inclination (i) of course is here smaller.

As the root surface of the gear contains the straightline element 240offset from its axis 32, it is a hyperboloid of revolution. The profile246 of an axial section of this root surface is concave. Theoreticallythe axial profile 239 of the outside surface of the pinion 231 shouldalso be concave, but only slightly so. A straight profile il: .111 1. 1J. may bessubstituted therefor. the. outside surface of the gear 230 maybe made straight, unlessxthergear. hasa small tooth number. In that caseit is preferably made convex.

Characteristics ,Themain characteristics will now be further describedwithFigures 12 to 18. While these correspond directly to the firstdescribed embodiment of Figures 1 to 8, they apply; also in a generalway to the other described emime t The teeth follow pitch surfaces,which are surfaces of revolution coaxial with the respective gear. Theside snrfaces of the teeth are warped surfaces whose profile inclinationto the normals of their pitch surfaces changes, generally by at leasttwenty degrees from end to end of the,teet h. U

.F igures 12 and 13 further illustrate the warped or twisted toothsurfaces, and their rapid change of profile inclination; Fig.- 14 showson the left the longitudinally convexside 48 of the gear teeth, and atthe. right the longitudinally concave side 49 thereof. Along the endprofile 76 of side 48 the surface normals coincide with The axialprofile 247 r,

eta-tar thegears riuna direction so that mesh starts at the outer, endsof theteeth, the, pitch line approach of the pinion to point53. of the.gear is in a relative path 55 to point 53, and the recess is in a path55'. Immediately thelimit normals,-and mating tooth surfaces have equalandmatching convex and concave curvature. Along the end profile 77 ofthe opposite side also the surface normals-coincide with the limitnormal, so that the same kindof very intimate contact results.

The intimacy of tooth contact remains high all along the, length of theteeth, although it decreases gradually. Even at the opposite end itcompares favorably with the tooth contact attained with conventionalteeth.

The, position of the end profiles 76, 77 of matched contact is furthershown in Fig. 15. In the upper part of this figure,'these profiles areat the actual ends of the teeth. The lower part has the same endprofiles 76, 77, withgsupporting portions 78, 80 added at opposite endsto increase the tooth strength.

..,.The axial section of the pinion 31, Fig. 17, shows belowtherlongitudinally concave tooth side 81, that meshes with side 48 ofthe gear. Its longitudinally convex tooth side-82 is shown above. Itmeshes with side 49 of the gear: The pinion should have an outsidesurface whose axialprofile is moderately concave, as indicated in Fig.3. :LA; conical surface 83 is shown substituted therefor, asrma'y bedone sometimes in practice. This may result instooth tops whose width issmallest in a region intermediate the ends of the teeth.

The pinion teeth 84 overlap the ends of the gear teeth. They do notterminate at the points 53, 54 (Fig. 1) but reachcbeyond. Pinion toothsurface 81 contains a line 85: that corresponds to the end profile 76 ofthe gear tooth-surfaces 48 and gets into gear contact with said profile.It is its mating line of the pinion. On the pinion tooth surface 82 is aline 86 mating with end profile 77 of the gear.

In accordance with the invention the pinion tooth surfaces 81 arerelieved adjacent end 50, starting at line 85. Andzthe pinion toothsurfaces 82 are relieved adjac'en't end 51',starting at line 86. Thisapplies to all described embodiments. The relief is best seen in Fig.18, and is also shown in Fig. 3.

--The'-relieved portion 87 is at an angle to surface 81. Likewise therelieved portion 88 is at an angle to surface 82. The said angles differfrom 180 degrees by l'ess' tha'n twelve degrees. It should be enough toprevent contact. at portions 87, 88 at any actual running condition.This is not a mere ease-off, but a deliberate destru'ction'of thesurface. The surfaces 81, 87 and 82, 88 meet in a ridge, which mayhowever be rounded. It may. be said that they meet in a near-ridge.

The reason for this important feature will now be described with Fig.16. This is a normal section through the teeth contacting at end point53, as if the teeth would continue. Let us consider the pitch linecontact. When adjacent point 53 thepaths 55, 55' are in the direction oftangent 56, inclined at the limit pressure angle to the vertical, ,aswas described for a mean section with Fig. 4. At point 53 this is alsothe inclination of the tooth surface, which is perpendicular to thelimit normal. In other words, this direction follows the tooth surfaceand is tangent thereto. The dotted portion 55 of the path seeminglyinterferes with the gear tooth. Although the gear tooth terminates at53, there would be actual interference if the tooth surface of thepinion would continue beyond this point without relief, because at thisend the contactingsurfaces are so closely matched as to have equalconvex and concave curvatures in all sections through point 53.

The invention prevents interference by using warped tooth surfaces andproviding working surfaces 81, 82 on the pinion that terminate at lines85, 86 respectively. The relieved remainder of the tooth length is forstrength, not for tooth action. e I

Fig. 16 further illustrates the; leaning teeth usually found at theouter end. One side profile of the teeth is negatively inclined, seealso Fig. 5.

Production A preferred form of production is by form-cutting bothmembers of the hyp'oid gear pair. Figures 19 and 20 show a pair .oftools, 90,.90' for rough-cutting opposite sides of the toothspaces 91shown in dotted lines- Preferably the workpiece is,continuously anduniformly rotated on its axis, while the tools 90,90 perform periodichelical reciprocations, each about its own axis (40). Preferably apluralityoftool pairs 90, are used. Their axes are angularly spacedabout the workpiece. The duration of the reciprocation is such that eachtool enters a different tooth space on successive strokes. Thus theworkpiece in effect is indexed from stroke to stroke through a numberofteeth. This number should be prime to the tooth number of the workpiece,so that each tool successively cuts inall toothspaces.

Tool 90 cuts with side 92 and, end 93. Tool 90' cuts with, theoppositeside 92, and end 93. Together they rough out the tooth spaces.There is also a clapping motion to keepthe tools clear of theworkpieceduring the return strokes. The depthwise feed and also the clappingmotion are preferably made up of an axial motion of the gear blank or,workpiece and of a turning motion about the gear axis, so that the feedpath about bisects the angle between opposite tooth sides at themid-section.

Fig. 14 shows at the left various positions of the describing line .(57)to be represented by a finish-cutting edge. It is seen that theinclination of the describing line to the tooth bottom 46 varies. Thisvariation presents a problem in cutting clearance, both in end clearanceand in side clearance. On roughing tools 98, 90" I may solve thisproblem by tilting each tool as it goes through the cut, so as to obtainmore nearly constant cutting clearances; The direction of the tilt axisismade up of a direction adapted to keep the end clearance right and ofa direction to keep the side clearance right. It is obtainable byvectorial addition. This axis-partakes in the helical. motion of thetool and lies on the slide member of said motion. Generally the tiltaxis is inclined to the axis (48) of the helical 'tool stroke andpreferably passes through the center 94 of the edge round of the tool.As the tool clearances change with changing depth of cut, I may changethe phase of the tilting motion with increasing depth of cutaccordingly.

A pair of tools are provided for finishing opposite sides of the teeth.Each of these tools cuts with one side only and isskept clear of thetooth bottom. One such tool is shown in Figures 21 to 23. Tool 95 has afinish-cutting 11 edge 96 formed at the intersection of a cutting face97 with a side surface 9 8. The tool is preferably designed for contourgrinding, to provide maximum cutting clearance. It is sharpened byregrinding a narrow strip 98 of side surface 98, that follows thecutting edge 96. After sharpening the tool is adjusted to advance thenewly formed cutting edge to the original position. This adjustment isin a direction parallel to the cutting face 97. The blade is alignedangularly with the cutting face, so that sidewise displacement providesthe required direction of adjustment. The tool is secured in diflerentsidewise positions by using parallels 99 of different thickness.

Fig. shows that the direction of a warped tooth surface in sectionsparallel to the drawing plane depends much on the depth level of thesection. Thus the different points of a finishing edge (96) cut indifferent directions. To attain improved side clearance on thecontourground finishing edge I vary the lateral inclination of strip 98'at different depths in accordance with the difference in cuttingdirection. This is illustrated in Fig. 23 by sections taken at differentlevels parallel to the drawing plane. For clarity the sectional linesare shown extended beyond the narrow strip 98. Cutting at mean point 41is in a direction 100. The sectionalprofile there is in a direction 102,while near the top of the cutting tooth it is in a direction 101 andnear the bottom it is in a direction 103. Although the shown cuttingedge is straight, the strip 98' is not a plane but a twisted surface ofvarying inclination. It is approximately a helical surface followingedge 96. Such strips of varying inclination may also be used with curvedcutting edges.

The finishing blades may be used in the same cutter assembly with themultiple pairs of roughing blades. In this case they are kept clear ofthe workpiece during the depth feed, and are advanced into cuttingposition when full cutting depth is reached. They may be advanced by aslight displacement in the direction of the axis of their helicalcutting motion.

To decrease the change of cutting clearance during the cutting strokes,a finishing blade may be tilted about the cutting edge as it goesthrough the cut.

The pinions may be finished like the gear, except that the clappingmotion is usually not along and about the work axis, but in a directionmore nearly perpendicular to the pinion pitch surface.

Figures 24 to 27 illustrate a roughing operation chiefly for pinions.Here four different roughing tools 105, 106, 107, 108 are provided, eachshown in one of the four figures. Tool 105 is fed in its helicallyreciprocating slide part to follow the tooth profile 110, starting froma position 105'. Tool 108, Fig. 27, starts from a position 108' and isfed to follow tooth profile 111. These tools are also tilted as they gothrough the cutting stroke, preferably about axes that pass through thecenters 94 of the edge rounds. These axes partake in the helical motionof the respective tools and in the feed motion. The intermediate tools106, 107 are similarly tilted about axes intersecting their end cuttingedge. Tool 106 is fed from a starting position 106, while tool 107starts from a position 107' in its helically reciprocating slide part.The four tools completely cover the entire width of the tooth space evenwhere it is widest. They work together, in support of each other,because each tool enters a ditferent tooth space in successive strokes.

The clapping of each tool is preferably in the direction of its feedingmotion. The tools 106, 107 are wider than the minimum slot width of thetooth space, and their feed is terminated in the position shown in fulllines. From here on the tools 105, 108 only of the four tools proceedwith the depth feed.

Tool clearance problems are minimized in the embodiment now to bedescribed with Figures 28 and 29. Here the warped tooth surface ofthegear is approximated by a helical surface of constant lead, such asmay be described by a cutting edge that moves helically along and aboutan axis while the workpiece stands still. Helix 119 is described by meanpoint 41 of the cutting edge so that it approximates the required pitchline. Its curvature and curvature plane coincides with the curvature andcurvature plane of the pitch line. pitch-line normal that lies in thecurvature plane. Curvature center 121 lies on this normal. The axis 122of the sought helical surface intersects normal at right angles at apoint 123 such that the required change of profile inclination isobtained lengthwise of the tooth as Well as the curvature required onhelix 119. Axis 122 is inclined to plane 124 at an angle smaller thansixty degrees. Plane 124 is the tangent plane of the pitch surface atmean point 41.

The workpiece is intermittently indexed to present other tooth spaces tothe cutting edge. The tooth sides of the pinion are here produced in agenerating operation, in which helically reciprocating cutting edgesdescribe tooth sides of the gear while thepinion meshes with and rolls onsaid gear. 7 v

In a further embodiment the tooth sides of the gear are described byhelically moving cutting edges while the gear turns on its axis at aslow enough rate that each cutting edge enters adjacent tooth spaces onsuccessive strokes. The pinion is here also produced by generationwherein cutting edges describe tooth sides of the rotating gear- Stillfurther modifications may be made. This application is intended to coverany variations, uses, oradaptations of the invention following, ingeneral, the principles of the invention and including such departuresfrom the present disclosure as come within the known or customarypractice in the art to which the invention pertains and as may beapplied to the essential features herein set forth and as fall withinthe scope of the invention or the limits of the appended claims.

I claim:

1. Hypoid gearing comprising a pair of gears having angularly disposedand offset axes, the teeth of each member of said pair following asurface of revolution coaxial with said member and their side surfacesbeing warped surfaces whose profile inclination to the normals of saidsurface of revolution changes along the length of the teeth anddecreases on one side of the teeth while increasing on the opposite sidefrom one tooth end to the other, the tooth surface normals at pointsadjacent one end of the teeth coinciding approximately with the limitnormals at said points, so that near-surface contact is achieved at saidpoints.

2. Hypoid gearing comprising a gear and a pinion adapted to mesh witheach other and having angularly disposed and offset axes, the teeth ofsaid gear following a surface of revolution coaxial with its axis andtheir side surfaces being warped surfaces whose profile inclination tothe normals of said surface of revolution changes along the length ofthe teeth and decreases on one side of the teeth while increasing on theopposite side from one tooth end to the other, the tooth surface normalsat points of one end profile of said gear teeth coinciding approximatelywith the limit normals at said points, so that near-surface contact isachieved at said points. i

3. Hypoid gearing according to claim 2, wherein the gear teeth arecurved lengthwise and extend between outer and inner ends, and whereinthe tooth surface normals at the points of the outer end profilecoincide with the limit normals at said points on the longitudinallyconvex side of the gear teeth, and wherein the tooth surface normals atthe points of the inner end profile coincide with the limit normals atsaid points on the longitudinally concave side of the gear teeth.

4. Hypoid gearing according to claim 2, wherein the teeth of the pinionoverlap the ends of the gear teeth lenghtwise, the tooth sides of saidpinion being relieved at the end that meshes with near-surface contact,relief Numeral 120 denotes the' starting at a line that corresponds toand meshes with the end profile of the gear teeth.

5. Hypoid gearing comprising a gear and a pinion adapted to mesh witheach other and having angularly disposed and offset axes, the teeth ofsaid gear being curved lengthwise and extending between outer and innerends along a surface of revolution coaxial with its axis, the sidesurfaces of said teeth being warped surfaces whose profile inclinationto the normals of said surface of revolution changes along the length ofthe teeth, decreasing from the outer end to the inner end on thelongitudinally concave side of the gear teeth while increasing on thelongitudinally convex side, the teeth of the pinion overlapping the endsof the gear teeth, the longitudinally concave tooth sides of the pinionthat mate with the longitudinally convex tooth sides of the gear beingrelieved adjacent their outer end, the opposite tooth sides of thepinion being relieved adjacent their inner end, relief starting with anear-ridge at a line of the pinion tooth surfaces that corresponds toand meshes with the outer and inner end profiles of the gearrespectively, so that interference is avoided in the region ofnear-surface contact.

6. Hypoid gearing according to claim 5, wherein the gear teeth areleaning at their outer end to such an extent that the longitudinallyconvex side is negatively inclined to the normals of the surface ofrevolution along which the teeth extend.

7. Hypoid gearing according to claim 2, wherein the gear and pinion haveapproximately straight profiles in a mid-section normal to the teeth,and wherein the teeth of the gear extend between end surfaces thatconverge from bottom to top of the gear, so that the face width issmaller at the tooth tops than at the bottoms.

8. Hypoid gearing comprising a gear and a pinion adapted to mesh witheach other and having angularly disposed and offset axes, the teeth ofsaid gear following a surface of revolution coaxial with its axis andextending between an outer and an inner end surface, the side surfacesof said gear teeth being warped surfaces whose profile inclination tothe normals of said surface of revolution changes from end to end ofsaid teeth by at least twenty degrees, decreasing on one side of theteeth and increasing on the opposite side, said end surfaces convergingfrom bottom to top of the gear teeth, and the pinion tooth surfacesbeing relieved at the end of lower profile inclination of the toothsides, relief starting with a near-ridge along a line that mates withthe end profile of the gear teeth.

9. Hypoid gearing having approximately complementary tooth profiles in amean section normal to the tooth direction, the teeth of one memberoverlapping the tooth ends of the other member of the gear pair, thetooth sides of said one member being relieved adjacent one end, reliefstarting with a near-ridge along a line that mates with the end profileof the other member.

10. Hypoid gearing according to claim 9, wherein the gearing consists ofa gear and a pinion, and wherein the gear contains convex tooth profilesand the pinion contains concave tooth profiles.

11. Hypoid gearing according to claim 1, wherein the 14 tooth sides ofat least one member of the gear pair are surfaces containing a constantline all along their working length, and are such as may be described onthe uniformly rotating one member by said line moving at a uniform ratein a helical path, the axis of said path being inclined to and offsetfrom the axis of said one member.

12. Hypoid gearing according to claim 11, wherein the mean direction ofthe describing line is offset from the axis of its helical path.

13. Hypoid gearing according to claim 11, wherein the describing line isa straight line whose extension is offset from the axis of its helicalpath.

14. Hypoid gearing according to claim 2, wherein the tooth sides of eachmember of the gear pair contain a constant line all along their workinglength, and are such as may be described on the respective uniformlyrotating member by said line performing a uniform helical motion ofconstant lead, approximately as if said line were rigid with a basichelical member of the hypoid gear pair.

15. A tapered gear having longitudinally curved teeth that extendbetween an outer end surface and an inner end surface and follow asurface of revolution coaxial with the gear, the side surfaces of saidteeth being warped surfaces whose profile inclination to the normals ofsaid surface of revolution changes by at least twenty degrees from theouter end of said teeth to their inner end, decreasing on thelongitudinally concave side of the gear teeth and increasing on thelongitudinally convex side, said end surfaces being surfaces ofrevolution converging from bottom to top of the teeth, so that the facewidth is smaller at the tooth tops than at the tooth bottoms.

16. A tapered gear according to claim 15, wherein the end surfacesconverge at an angle of at least thirty degrees and wherein the toothtops lie in a surface of revolution of convex profile in axial section.

17. A tapered pinion having spirally arranged and longitudinally curvedteeth extending along a surface of revolution coaxial with said pinionbetween an outer end and an inner end, the side surfaces of said teethbeing warped surfaces whose profile inclination to the normals of saidsurface of revolution changes by at least twenty degrees from the outerend of said teeth to their inner end, decreasing on the longitudinallyconvex side of the teeth and increasing on the longitudinally concavesides, the working portions of said side surfaces terminating at one endof the teeth with a near-ridge followed by a relieved portion angularlyinclined to said working portion at an angle differing from degrees byless than twelve degrees, said near-ridge and relieved portion beingadjacent the inner and outer tooth ends on the longitudinally convex andconcave tooth sides respectively.

References Cited in the file of this patent UNITED STATES PATENTS2,105,104 Wildhaber et al Ian. 11, 1938 2,174,814 Ackerman Oct. 3, 19392,183,285 Wildhaber Dec. 12, 1939 2,302,942 Golber Nov. 24, 19422,358,489 Carlsen Sept. 19, 1944 2,374,890 Pelphrey May 1, 1945

